Some of the commenters (most notably and intelligibly Blackadder) have noted that Rothbard's view here - that one cannot voluntarily sell themselves into slavery - seems contradictory...to the axiom of absolute self-ownership presumably. I can't pretend to speak for Rothbard, and surely there are plenty of people more knowledgeable on his views than I, but I believe such outright criticism may be somewhat unfounded. I find Rothbard's views here to be somewhat more subtle than what has been presented.
It seems, in reading Rothbard, one gets the sense that one of the core features of property is its alienability - something by which we may part ownership and control of and of which through transfer of title may be adopted by others. It's not a term he uses very much outside of the discussion on slavery. But I don't think it's because he's trying to make a clever exception to his theory on property when it's being put to the test. I think this qualification applies across the board and is in fact consistent with the rest of his theory; he simply need not invoke such aspects when they are not of relevance.
So we can see with something like a rock, we may easily part our will, control, and ownership of it and cede these rights to others. But Rothbard contends that it's not quite as easy to do such with a human being, because our will is inalienable and cannot (at least not yet) be separated from us, materially without separating (parts of) our bodies, in some real sense, from ourselves. In other words, I can contract out and sell positive obligations to people (employment, etc.), but since I cannot divorce my will from my physical self, and thus cannot relinquish physical control in a literal way, I cannot, in any meaningful sense, parcel out ownership of that which my will still embodies. As such, all such obligations, in this sense, can be "backed out of" (and compensation may follow based on the contract at hand). And so, Rothbard's view is not that de facto chattel slavery could not or did not exist. Rather, it would have been his belief that involuntary slavery is merely coercion masquerading as title exchange, while voluntary slavery is contractual employment masquerading as title exchange.
Now, the interesting consequence of such a view - and it's not one that Rothbard delves into, to my knowledge - is that it doesn't preclude the voluntary severing of your will from your body in any meaningful sense. For instance, it's perfectly fine, as long as the action remains voluntary until completion, to let someone cut off your hair and then claim ownership of it. To push it further, you could take blood, an organ, a limb, and even the head itself. I'm not trying to be too graphic here - I just want to illustrate the point. Those things would be fine, so long as they were voluntary, under the Rothbardian view. And once those things had been literally separated from your will (or control) then they could quite literally be owned. Likewise it might be permissible, if not disrespectful, under Rothbardian norms to homestead a corpse (I smell a zombie movie!).
I'm making some of these points not to show you what creeps us Rothbardians can be, but rather to illustrate two main points. The first is that there is no mystical holy sanction or reverence that Rothbard gives to the human body per se. The second is that his views aren't saturated with the inconsistencies that some of his nominal detractors might think they are. He simply draws a line in the philosophical sand stating that if you literally cannot relinquish control over something in a very literal sense, then you simply cannot transfer ownership of it in any meaningful way.